Question

The region bounded by the given curves is rotated about the specified axis. find the volume v of the resulting solid by any method. y = −x2 + 9x − 20, y = 0; about the x-axis

Answer 1

Answer: $V = \frac{\pi}{30}$

Explanation:

First, we compute the points of intersection of the curves $y = -x^2 + 9x - 20$ and $y = 0$. Since the equation have y as their left sides, 

$-x^2 + 9x - 20 = 0 \\-(x - 5)(x - 4) = 0 \\ x = 5, x = 4$

So, the curves intersect at x = 5 and x = 4. Using the ring method

$V = \int_{4}^{5}{\pi(-x^2 + 9x - 20)^2}dx$     (1)

In the ring method, if the region is rotated about x-axis, we use dx in the integration and the independent variable is x. Because the independent variable is x, we use the x-coordinates of points of intersection as the limits of integration.

Note that in equation (1), we use 4 and 5 as limits of integration because the given curves $y = -x^2 + 9x - 20$ and y =0 intersect at x = 4 and x = 5.

Hence, using equation (1), the volume of the solid is given by:

$V = \int_{4}^{5}{\pi(-x^2 + 9x - 20)^2}dx \\ \indent = \pi\int_{4}^{5}{(x^4 -18x^3 + 121x^2 − 360x + 400)}dx \\ \indent = \pi\left [ \frac{x^5}{5} - \frac{9x^4}{2} + \frac{121x^3}{3} - 180x^2 + 400x + C \right ]_{4}^{5} \\ \indent \boxed{V = \frac{\pi}{30}}$